Abstract
To what extent does Noether's principle apply to quantum channels? Here, we quantify the degree to which imposing a symmetry constraint on quantum channels implies a conservation law, and show that this relates to physically impossible transformations in quantum theory, such as time-reversal and spin-inversion. In this analysis, the convex structure and extremal points of the set of quantum channels symmetric under the action of a Lie group $G$ becomes essential. It allows us to derive bounds on the deviation from conservation laws under any symmetric quantum channel in terms of the deviation from closed dynamics as measured by the unitarity of the channel. In particular, we investigate in detail the $U(1)$ and $SU(2)$ symmetries related to energy and angular momentum conservation laws. In the latter case, we provide fundamental limits on how much a spin-$j_A$ system can be used to polarise a larger spin-$j_B$ system, and on how much one can invert spin polarisation using a rotationally-symmetric operation. Finally, we also establish novel links between unitarity, complementary channels and purity that are of independent interest.
Highlights
These facts imply that a complex disconnect occurs between symmetries of a system and traditional conservation laws when we extend the analysis to open dynamics described by quantum channels; see Fig. 1
The time-reversal transformation t → −t is a stark example of a symmetry transformation that does not correspond to any physical transformation that could be performed on a quantum system A [9]
We extend previous results on the structure of symmetric channels [12,13,14,15,16] and derive novel relations for the unitarity of a quantum channel [17], both of which are of independent interest to the quantum information community
Summary
Noether’s theorem in classical mechanics states that for every continuous symmetry of a system there is an associated conserved charge [1,2,3]. The most general kind of evolution of a quantum state, for relativistic or nonrelativistic quantum theory, is not unitary dynamics but instead a quantum channel. Given a symmetry principle, there exist quantum channels that can change the expectation of the generators of the symmetry in nontrivial ways These facts imply that a complex disconnect occurs between symmetries of a system and traditional conservation laws when we extend the analysis to open dynamics described by quantum channels; see Fig. 1. Given this breakdown of Noether’s principle, our primary aim in this work is to address the following fundamental question: Q1. We shall see that this question relates to the distinction between the notion of an active transformation and a passive transformation of a quantum system
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